Relative projectivity and relative endotrivial modules.
type: Item
instance of: scholarly article
title: Relative projectivity and relative endotrivial modules.
zbMATH Open document ID: 1257.20003
author: Caroline Lassueur
published in: Journal of Algebra
publication date: 2012-01-11T00:00:00Z
full work available at URL: http://infoscience.epfl.ch/record/161952
review text: Let \(G\) be a finite group, \(k\) an algebraically closed field of characteristic \(p>0\) and \(V\) a finitely generated \(kG\)-module. A finitely generated \(kG\)-module \(M\) is called relatively \(V\)-projective if \(M\) is a direct summand of \(V\otimes N\) for some finitely generated \(kG\)-module \(N\). This notion generalizes the relative projectivity with respect to subgroups. Now suppose that \(V\) is absolutely \(p\)-divisible, i.e. \(p\) divides \(\dim U\) for every direct summand \(U\) of \(V\). (Otherwise every finitely generated \(kG\)-module is relatively \(V\)-projective.) The author calls a finitely generated \(kG\)-module \(M\) relatively \(V\)-endotrivial if \(\Hom_k(M,M)\cong k\oplus P\) where \(P\) is a relatively \(V\)-projective \(kG\)-module. This notion generalizes the classical notion of an endotrivial \(kG\)-module. The author investigates the behavior of relatively endotrivial modules under restriction, induction, inflation, the Green correspondence and the Heller functor. Every relatively \(V\)-endotrivial \(kG\)-module \(M\) has a unique indecomposable direct summand \(M_0\) which is also relatively \(V\)-endotrivial. Two relatively \(V\)-endotrivial \(kG\)-modules \(M,N\) are called equivalent if \(M_0\) and \(N_0\) are isomorphic. The tensor product induces the structure of an Abelian group on the set \(T_V(G)\) of equivalence classes of relatively \(V\)-endotrivial \(kG\)-modules. The author studies the behavior of \(T_V(G)\) under restriction and inflation. In the case where \(G\) is a finite \(p\)-group, the Dade group \(D(G)\) of capped endo-permutation \(kG\)-modules can be embedded into \(T_{\widetilde V}(G)\), for \(\widetilde V:=\bigoplus_{H<G}\text{Ind}_H^G(k)\). The author raises the question whether \(D(G)\cong T_{\widetilde V}(G)\). In the last part of the paper she deals with the situation where \(G\) is a finite group and the Sylow \(p\)-subgroups of \(G\) are cyclic or of order \(4\).
reviewed by: Burkhard Külshammer
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 19A22
zbMATH DE Number: 5996443
zbMATH Keywords: finite groups
zbMATH Keywords: finitely generated modules
zbMATH Keywords: endotrivial modules
zbMATH Keywords: relative projectivity
zbMATH Keywords: Dade groups
zbMATH Keywords: stable categories
zbMATH Keywords: endo-permutation modules
zbMATH Keywords: Heller translate
zbMATH Keywords: Green correspondence
zbMATH Keywords: restriction
zbMATH Keywords: induction
zbMATH Keywords: inflation
MaRDI profile type: Publication
OpenAlex ID: W2133980464
cites work: Endotrivial modules in the cyclic case.
cites work: Q3895647
cites work: Relative projectivity and ideals in cohomology rings
cites work: Transfer maps and virtual projectivity
cites work: Q5648522
cites work: Almost-split sequences and group rings
cites work: Q3996909
cites work: Nilpotent elements in the Green ring
cites work: Q4399851
cites work: Tensor induction of relative syzygies
cites work: The group of endo-permutation modules
cites work: Endotrivial modules for finite groups of Lie type
cites work: The group of endotrivial modules in the normal case.
cites work: Endo-permutation modules over p-groups. I
cites work: Endo-permutation modules over p-groups. II
cites work: Certain representation algebras
cites work: Q4846455
DOI: 10.1016/J.JALGEBRA.2011.04.003
description: scientific article; zbMATH DE number 5996443